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The Role of Nonassociative Algebra in Projective Geometry
John R. Faulkner, University of Virginia, Charlottesville, VA
 Graduate Studies in Mathematics 2014; 229 pp; hardcover Volume: 159 ISBN-10: 1-4704-1849-5 ISBN-13: 978-1-4704-1849-6 List Price: US$67 Member Price: US$53.60 Order Code: GSM/159 Not yet published.Expected publication date is October 23, 2014. See also: Graduate Algebra: Noncommutative View - Louis Halle Rowen Parabolic Geometries I: Background and General Theory - Andreas Cap and Jan Slovak There is a particular fascination when two apparently disjoint areas of mathematics turn out to have a meaningful connection to each other. The main goal of this book is to provide a largely self-contained, in-depth account of the linkage between nonassociative algebra and projective planes, with particular emphasis on octonion planes. There are several new results and many, if not most, of the proofs are new. The development should be accessible to most graduate students and should give them introductions to two areas which are often referenced but not often taught. On the geometric side, the book introduces coordinates in projective planes and relates coordinate properties to transitivity properties of certain automorphisms and to configuration conditions. It also classifies higher-dimensional geometries and determines their automorphisms. The exceptional octonion plane is studied in detail in a geometric context that allows nondivision coordinates. An axiomatic version of that context is also provided. Finally, some connections of nonassociative algebra to other geometries, including buildings, are outlined. On the algebraic side, basic properties of alternative algebras are derived, including the classification of alternative division rings. As tools for the study of the geometries, an axiomatic development of dimension, the basics of quadratic forms, a treatment of homogeneous maps and their polarizations, and a study of norm forms on hermitian matrices over composition algebras are included. Readership Graduate students and research mathematicians interested in nonassociative algebra and projective geometry; physicists interested in division algebras and string theory.